Quotient vs Ratio vs Fraction vs Rational

Question

I can't seem to differentiate between a Quotient and a Ratio and a Fraction and a Rational. From what I know a rational is a number like $2/3$ or $5.4/7$ whereas quotient, ratio and fraction all are just three names for one same thing $1+x/1-x^3$ , and that is just same as rational number except that a variable is used instead of a number. Am I correct ?

Answer 1

Not exactly.

A quotient is used for the result of dividing an integer into another. It is also used in similar or analogous senses -- anywhere one reduces an object by another, the result of this reduction is called a quotient -- e.g., the set of equivalence classes of a set with respect to an equivalence relation.

The words ratio and fraction are more alike, although ratio carries a more analytic flavour while fraction sounds algebraic -- at any rate the two stand for any expressions of the forms $A:B,A/B$ and $AB^{-1},$ without regard to whatever meaning they could have. In short, these words (fraction is the more frequently used these days) refer to a merely formal expression.

Finally, a rational number, or a fraction of integers, is simply a ratio or fraction where $A$ and $B$ are both integers, with $B\ne0,$ of course.

Answer 2

The easiest of these to define is a rational number. It's a number that can be written as a fraction with integer numerator and denominator, even if it doesn't visibly appear that way. So these are all rational numbers (in fact, the same rational number): $$ \frac{3}{2} , \ 1.5, \ 1 + \frac{1}{2},\ \frac{1}{2/3}, \ \frac{3\pi}{2\pi} \ . $$

A quotient is an expression formed by dividing one thing by another. So these are quotients: $$ \frac{3}{2}, \frac{3\pi}{2}, \ \frac{3+ \pi}{2}, \frac{x+ 5}{x^2} \ . $$

Sometimes you think of a quotient expression as the number it represents, when it happens to represent a number. Sometimes that number is rational, sometimes it isn't.

A ratio is a little more subtle. It usually occurs in an application of some kind. For example, if the ratio of boys to girls in a math class is $2:3$ then $2/5$ of the class are boys. If you are travelling at $20$ miles per hour then the ratio of distance covered to hours spent is $20:1$.

Answer 3

$$\frac AB$$ is a ratio, whatever $A$ and $B$ denote. Fraction is a quasi synonym, that also conveys the graphical representation with a division bar.

A quotient is the result of a division, especially when there can be a remainder (as in integer and polynomial divisions).

A rational (number) is any number that can be expressed as the ratio of two integers.

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Addendum:

A rational fraction is the ratio of two polynomials.